fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
inv_mem_connectedComponent_one {G : Type*} [TopologicalSpace G] [DivisionMonoid G]
[ContinuousInv G] {g : G} (hg : g ∈ connectedComponent (1 : G)) :
g⁻¹ ∈ connectedComponent (1 : G) := by
rw [← inv_one]
exact
Continuous.image_connectedComponent_subset continuous_inv _
((Set.mem_image _ _ _).mp ⟨g, hg, rfl⟩) | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | inv_mem_connectedComponent_one | null |
@[to_additive /-- The connected component of 0 is a subgroup of `G`. -/]
Subgroup.connectedComponentOfOne (G : Type*) [TopologicalSpace G] [Group G]
[IsTopologicalGroup G] : Subgroup G where
carrier := connectedComponent (1 : G)
one_mem' := mem_connectedComponent
mul_mem' hg hh := mul_mem_connectedComponent_one hg hh
inv_mem' hg := inv_mem_connectedComponent_one hg | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.connectedComponentOfOne | The connected component of 1 is a subgroup of `G`. |
@[to_additive
/-- An additive monoid homomorphism (a bundled morphism of a type that implements
`AddMonoidHomClass`) from an additive topological group to an additive topological monoid is
continuous provided that it is continuous at zero. See also
`uniformContinuous_of_continuousAt_zero`. -/]
continuous_of_continuousAt_one {M hom : Type*} [MulOneClass M] [TopologicalSpace M]
[ContinuousMul M] [FunLike hom G M] [MonoidHomClass hom G M] (f : hom)
(hf : ContinuousAt f 1) :
Continuous f :=
continuous_iff_continuousAt.2 fun x => by
simpa only [ContinuousAt, ← map_mul_left_nhds_one x, tendsto_map'_iff, Function.comp_def,
map_mul, map_one, mul_one] using hf.tendsto.const_mul (f x)
@[to_additive continuous_of_continuousAt_zero₂] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_of_continuousAt_one | If a subgroup of a topological group is commutative, then so is its topological closure.
See note [reducible non-instances]. -/
@[to_additive
/-- If a subgroup of an additive topological group is commutative, then so is its
topological closure.
See note [reducible non-instances]. -/]
abbrev Subgroup.commGroupTopologicalClosure [T2Space G] (s : Subgroup G)
(hs : ∀ x y : s, x * y = y * x) : CommGroup s.topologicalClosure :=
{ s.topologicalClosure.toGroup, s.toSubmonoid.commMonoidTopologicalClosure hs with }
variable (G) in
@[to_additive]
lemma Subgroup.coe_topologicalClosure_bot :
((⊥ : Subgroup G).topologicalClosure : Set G) = _root_.closure ({1} : Set G) := by simp
@[to_additive exists_nhds_half_neg]
theorem exists_nhds_split_inv {s : Set G} (hs : s ∈ 𝓝 (1 : G)) :
∃ V ∈ 𝓝 (1 : G), ∀ v ∈ V, ∀ w ∈ V, v / w ∈ s := by
have : (fun p : G × G => p.1 * p.2⁻¹) ⁻¹' s ∈ 𝓝 ((1, 1) : G × G) :=
continuousAt_fst.mul continuousAt_snd.inv (by simpa)
simpa only [div_eq_mul_inv, nhds_prod_eq, mem_prod_self_iff, prod_subset_iff, mem_preimage] using
this
@[to_additive]
theorem nhds_translation_mul_inv (x : G) : comap (· * x⁻¹) (𝓝 1) = 𝓝 x :=
((Homeomorph.mulRight x⁻¹).comap_nhds_eq 1).trans <| show 𝓝 (1 * x⁻¹⁻¹) = 𝓝 x by simp
@[to_additive (attr := simp)]
theorem map_mul_left_nhds (x y : G) : map (x * ·) (𝓝 y) = 𝓝 (x * y) :=
(Homeomorph.mulLeft x).map_nhds_eq y
@[to_additive]
theorem map_mul_left_nhds_one (x : G) : map (x * ·) (𝓝 1) = 𝓝 x := by simp
@[to_additive (attr := simp)]
theorem map_mul_right_nhds (x y : G) : map (· * x) (𝓝 y) = 𝓝 (y * x) :=
(Homeomorph.mulRight x).map_nhds_eq y
@[to_additive]
theorem map_mul_right_nhds_one (x : G) : map (· * x) (𝓝 1) = 𝓝 x := by simp
@[to_additive]
theorem Filter.HasBasis.nhds_of_one {ι : Sort*} {p : ι → Prop} {s : ι → Set G}
(hb : HasBasis (𝓝 1 : Filter G) p s) (x : G) :
HasBasis (𝓝 x) p fun i => { y | y / x ∈ s i } := by
rw [← nhds_translation_mul_inv]
simp_rw [div_eq_mul_inv]
exact hb.comap _
@[to_additive]
theorem mem_closure_iff_nhds_one {x : G} {s : Set G} :
x ∈ closure s ↔ ∀ U ∈ (𝓝 1 : Filter G), ∃ y ∈ s, y / x ∈ U := by
rw [mem_closure_iff_nhds_basis ((𝓝 1 : Filter G).basis_sets.nhds_of_one x)]
simp_rw [Set.mem_setOf, id]
/-- A monoid homomorphism (a bundled morphism of a type that implements `MonoidHomClass`) from a
topological group to a topological monoid is continuous provided that it is continuous at one. See
also `uniformContinuous_of_continuousAt_one`. |
continuous_of_continuousAt_one₂ {H M : Type*} [CommMonoid M] [TopologicalSpace M]
[ContinuousMul M] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] (f : G →* H →* M)
(hf : ContinuousAt (fun x : G × H ↦ f x.1 x.2) (1, 1))
(hl : ∀ x, ContinuousAt (f x) 1) (hr : ∀ y, ContinuousAt (f · y) 1) :
Continuous (fun x : G × H ↦ f x.1 x.2) := continuous_iff_continuousAt.2 fun (x, y) => by
simp only [ContinuousAt, nhds_prod_eq, ← map_mul_left_nhds_one x, ← map_mul_left_nhds_one y,
prod_map_map_eq, tendsto_map'_iff, Function.comp_def, map_mul, MonoidHom.mul_apply] at *
refine ((tendsto_const_nhds.mul ((hr y).comp tendsto_fst)).mul
(((hl x).comp tendsto_snd).mul hf)).mono_right (le_of_eq ?_)
simp only [map_one, mul_one, MonoidHom.one_apply]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_of_continuousAt_one₂ | null |
IsTopologicalGroup.isInducing_iff_nhds_one
{H : Type*} [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {F : Type*}
[FunLike F G H] [MonoidHomClass F G H] {f : F} :
Topology.IsInducing f ↔ 𝓝 (1 : G) = (𝓝 (1 : H)).comap f := by
rw [Topology.isInducing_iff_nhds]
refine ⟨(map_one f ▸ · 1), fun hf x ↦ ?_⟩
rw [← nhds_translation_mul_inv, ← nhds_translation_mul_inv (f x), Filter.comap_comap, hf,
Filter.comap_comap]
congr 1
ext; simp
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.isInducing_iff_nhds_one | null |
IsTopologicalGroup.isOpenMap_iff_nhds_one
{H : Type*} [Monoid H] [TopologicalSpace H] [ContinuousConstSMul H H]
{F : Type*} [FunLike F G H] [MonoidHomClass F G H] {f : F} :
IsOpenMap f ↔ 𝓝 1 ≤ .map f (𝓝 1) := by
refine ⟨fun H ↦ map_one f ▸ H.nhds_le 1, fun h ↦ IsOpenMap.of_nhds_le fun x ↦ ?_⟩
have : Filter.map (f x * ·) (𝓝 1) = 𝓝 (f x) := by
simpa [-Homeomorph.map_nhds_eq, Units.smul_def] using
(Homeomorph.smul ((toUnits x).map (MonoidHomClass.toMonoidHom f))).map_nhds_eq (1 : H)
rw [← map_mul_left_nhds_one x, Filter.map_map, Function.comp_def, ← this]
refine (Filter.map_mono h).trans ?_
simp [Function.comp_def]
@[deprecated (since := "2025-09-16")]
alias TopologicalGroup.isOpenMap_iff_nhds_one := IsTopologicalGroup.isOpenMap_iff_nhds_one
@[deprecated (since := "2025-09-16")]
alias TopologicalGroup.isOpenMap_iff_nhds_zero := IsTopologicalAddGroup.isOpenMap_iff_nhds_zero | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.isOpenMap_iff_nhds_one | null |
@[to_additive /-- Let `A` and `B` be topological additive groups, and let `φ : A → B` be a
continuous surjective additive group homomorphism. Assume furthermore that `φ` is a quotient map
(i.e., `V ⊆ B` is open iff `φ⁻¹ V` is open). Then `φ` is an open quotient map, and in particular an
open map. -/]
MonoidHom.isOpenQuotientMap_of_isQuotientMap {A : Type*} [Group A]
[TopologicalSpace A] [ContinuousMul A] {B : Type*} [Group B] [TopologicalSpace B]
{F : Type*} [FunLike F A B] [MonoidHomClass F A B] {φ : F}
(hφ : IsQuotientMap φ) : IsOpenQuotientMap φ where
surjective := hφ.surjective
continuous := hφ.continuous
isOpenMap := by
intro U hU
rw [← hφ.isOpen_preimage]
suffices ⇑φ ⁻¹' (⇑φ '' U) = ⋃ k ∈ ker (φ : A →* B), (fun x ↦ x * k) ⁻¹' U by
exact this ▸ isOpen_biUnion (fun k _ ↦ Continuous.isOpen_preimage (by fun_prop) _ hU)
ext x
constructor
· rintro ⟨y, hyU, hyx⟩
apply Set.mem_iUnion_of_mem (x⁻¹ * y)
simp_all
· rintro ⟨_, ⟨k, rfl⟩, _, ⟨(hk : φ k = 1), rfl⟩, hx⟩
use x * k, hx
rw [map_mul, hk, mul_one]
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | MonoidHom.isOpenQuotientMap_of_isQuotientMap | Let `A` and `B` be topological groups, and let `φ : A → B` be a continuous surjective group
homomorphism. Assume furthermore that `φ` is a quotient map (i.e., `V ⊆ B`
is open iff `φ⁻¹ V` is open). Then `φ` is an open quotient map, and in particular an open map. |
IsTopologicalGroup.ext {G : Type*} [Group G] {t t' : TopologicalSpace G}
(tg : @IsTopologicalGroup G t _) (tg' : @IsTopologicalGroup G t' _)
(h : @nhds G t 1 = @nhds G t' 1) : t = t' :=
TopologicalSpace.ext_nhds fun x ↦ by
rw [← @nhds_translation_mul_inv G t _ _ x, ← @nhds_translation_mul_inv G t' _ _ x, ← h]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.ext | null |
IsTopologicalGroup.ext_iff {G : Type*} [Group G] {t t' : TopologicalSpace G}
(tg : @IsTopologicalGroup G t _) (tg' : @IsTopologicalGroup G t' _) :
t = t' ↔ @nhds G t 1 = @nhds G t' 1 :=
⟨fun h => h ▸ rfl, tg.ext tg'⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.ext_iff | null |
ContinuousInv.of_nhds_one {G : Type*} [Group G] [TopologicalSpace G]
(hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (fun x : G => x₀ * x) (𝓝 1))
(hconj : ∀ x₀ : G, Tendsto (fun x : G => x₀ * x * x₀⁻¹) (𝓝 1) (𝓝 1)) : ContinuousInv G := by
refine ⟨continuous_iff_continuousAt.2 fun x₀ => ?_⟩
have : Tendsto (fun x => x₀⁻¹ * (x₀ * x⁻¹ * x₀⁻¹)) (𝓝 1) (map (x₀⁻¹ * ·) (𝓝 1)) :=
(tendsto_map.comp <| hconj x₀).comp hinv
simpa only [ContinuousAt, hleft x₀, hleft x₀⁻¹, tendsto_map'_iff, Function.comp_def, mul_assoc,
mul_inv_rev, inv_mul_cancel_left] using this
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | ContinuousInv.of_nhds_one | null |
IsTopologicalGroup.of_nhds_one' {G : Type u} [Group G] [TopologicalSpace G]
(hmul : Tendsto (uncurry ((· * ·) : G → G → G)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : G, 𝓝 x₀ = map (fun x => x * x₀) (𝓝 1)) : IsTopologicalGroup G :=
{ toContinuousMul := ContinuousMul.of_nhds_one hmul hleft hright
toContinuousInv :=
ContinuousInv.of_nhds_one hinv hleft fun x₀ =>
le_of_eq
(by
rw [show (fun x => x₀ * x * x₀⁻¹) = (fun x => x * x₀⁻¹) ∘ fun x => x₀ * x from rfl, ←
map_map, ← hleft, hright, map_map]
simp) }
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.of_nhds_one' | null |
IsTopologicalGroup.of_nhds_one {G : Type u} [Group G] [TopologicalSpace G]
(hmul : Tendsto (uncurry ((· * ·) : G → G → G)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (x₀ * ·) (𝓝 1))
(hconj : ∀ x₀ : G, Tendsto (x₀ * · * x₀⁻¹) (𝓝 1) (𝓝 1)) : IsTopologicalGroup G := by
refine IsTopologicalGroup.of_nhds_one' hmul hinv hleft fun x₀ => ?_
replace hconj : ∀ x₀ : G, map (x₀ * · * x₀⁻¹) (𝓝 1) = 𝓝 1 :=
fun x₀ => map_eq_of_inverse (x₀⁻¹ * · * x₀⁻¹⁻¹) (by ext; simp [mul_assoc]) (hconj _) (hconj _)
rw [← hconj x₀]
simpa [Function.comp_def] using hleft _
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.of_nhds_one | null |
IsTopologicalGroup.of_comm_of_nhds_one {G : Type u} [CommGroup G] [TopologicalSpace G]
(hmul : Tendsto (uncurry ((· * ·) : G → G → G)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (x₀ * ·) (𝓝 1)) : IsTopologicalGroup G :=
IsTopologicalGroup.of_nhds_one hmul hinv hleft (by simpa using tendsto_id)
variable (G) in | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.of_comm_of_nhds_one | null |
@[to_additive
/-- Any first countable topological additive group has an antitone neighborhood basis
`u : ℕ → set G` for which `u (n + 1) + u (n + 1) ⊆ u n`.
The existence of such a neighborhood basis is a key tool for `QuotientAddGroup.completeSpace` -/]
IsTopologicalGroup.exists_antitone_basis_nhds_one [FirstCountableTopology G] :
∃ u : ℕ → Set G, (𝓝 1).HasAntitoneBasis u ∧ ∀ n, u (n + 1) * u (n + 1) ⊆ u n := by
rcases (𝓝 (1 : G)).exists_antitone_basis with ⟨u, hu, u_anti⟩
have :=
((hu.prod_nhds hu).tendsto_iff hu).mp
(by simpa only [mul_one] using continuous_mul.tendsto ((1, 1) : G × G))
simp only [and_self_iff, mem_prod, and_imp, Prod.forall, Prod.exists,
forall_true_left] at this
have event_mul : ∀ n : ℕ, ∀ᶠ m in atTop, u m * u m ⊆ u n := by
intro n
rcases this n with ⟨j, k, -, h⟩
refine atTop_basis.eventually_iff.mpr ⟨max j k, True.intro, fun m hm => ?_⟩
rintro - ⟨a, ha, b, hb, rfl⟩
exact h a b (u_anti ((le_max_left _ _).trans hm) ha) (u_anti ((le_max_right _ _).trans hm) hb)
obtain ⟨φ, -, hφ, φ_anti_basis⟩ := HasAntitoneBasis.subbasis_with_rel ⟨hu, u_anti⟩ event_mul
exact ⟨u ∘ φ, φ_anti_basis, fun n => hφ n.lt_succ_self⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.exists_antitone_basis_nhds_one | Any first countable topological group has an antitone neighborhood basis `u : ℕ → Set G` for
which `(u (n + 1)) ^ 2 ⊆ u n`. The existence of such a neighborhood basis is a key tool for
`QuotientGroup.completeSpace` |
@[to_additive const_sub]
Filter.Tendsto.const_div' (b : G) {c : G} {f : α → G} {l : Filter α}
(h : Tendsto f l (𝓝 c)) : Tendsto (fun k : α => b / f k) l (𝓝 (b / c)) :=
tendsto_const_nhds.div' h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.Tendsto.const_div' | null |
Filter.tendsto_const_div_iff {G : Type*} [CommGroup G] [TopologicalSpace G] [ContinuousDiv G]
(b : G) {c : G} {f : α → G} {l : Filter α} :
Tendsto (fun k : α ↦ b / f k) l (𝓝 (b / c)) ↔ Tendsto f l (𝓝 c) := by
refine ⟨fun h ↦ ?_, Filter.Tendsto.const_div' b⟩
convert h.const_div' b with k <;> rw [div_div_cancel]
@[to_additive sub_const] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.tendsto_const_div_iff | null |
Filter.Tendsto.div_const' {c : G} {f : α → G} {l : Filter α} (h : Tendsto f l (𝓝 c))
(b : G) : Tendsto (f · / b) l (𝓝 (c / b)) :=
h.div' tendsto_const_nhds | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.Tendsto.div_const' | null |
Filter.tendsto_div_const_iff {G : Type*}
[CommGroupWithZero G] [TopologicalSpace G] [ContinuousDiv G]
{b : G} (hb : b ≠ 0) {c : G} {f : α → G} {l : Filter α} :
Tendsto (f · / b) l (𝓝 (c / b)) ↔ Tendsto f l (𝓝 c) := by
refine ⟨fun h ↦ ?_, fun h ↦ Filter.Tendsto.div_const' h b⟩
convert h.div_const' b⁻¹ with k <;> rw [div_div, mul_inv_cancel₀ hb, div_one] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.tendsto_div_const_iff | null |
Filter.tendsto_sub_const_iff {G : Type*}
[AddCommGroup G] [TopologicalSpace G] [ContinuousSub G]
(b : G) {c : G} {f : α → G} {l : Filter α} :
Tendsto (f · - b) l (𝓝 (c - b)) ↔ Tendsto f l (𝓝 c) := by
refine ⟨fun h ↦ ?_, fun h ↦ Filter.Tendsto.sub_const h b⟩
convert h.sub_const (-b) with k <;> rw [sub_sub, ← sub_eq_add_neg, sub_self, sub_zero]
variable [TopologicalSpace α] {f g : α → G} {s : Set α} {x : α}
@[to_additive (attr := continuity) continuous_sub_left] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.tendsto_sub_const_iff | null |
continuous_div_left' (a : G) : Continuous (a / ·) := continuous_const.div' continuous_id
@[to_additive (attr := continuity) continuous_sub_right] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_div_left' | null |
continuous_div_right' (a : G) : Continuous (· / a) := continuous_id.div' continuous_const | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_div_right' | null |
@[to_additive (attr := simps! +simpRhs)
/-- A version of `Homeomorph.addLeft a (-b)` that is defeq to `a - b`. -/]
Homeomorph.divLeft (x : G) : G ≃ₜ G :=
{ Equiv.divLeft x with
continuous_toFun := continuous_const.div' continuous_id
continuous_invFun := continuous_inv.mul continuous_const }
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.divLeft | A version of `Homeomorph.mulLeft a b⁻¹` that is defeq to `a / b`. |
isOpenMap_div_left (a : G) : IsOpenMap (a / ·) :=
(Homeomorph.divLeft _).isOpenMap
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isOpenMap_div_left | null |
isClosedMap_div_left (a : G) : IsClosedMap (a / ·) :=
(Homeomorph.divLeft _).isClosedMap | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosedMap_div_left | null |
@[to_additive (attr := simps! +simpRhs)
/-- A version of `Homeomorph.addRight (-a) b` that is defeq to `b - a`. -/]
Homeomorph.divRight (x : G) : G ≃ₜ G :=
{ Equiv.divRight x with
continuous_toFun := continuous_id.div' continuous_const
continuous_invFun := continuous_id.mul continuous_const }
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.divRight | A version of `Homeomorph.mulRight a⁻¹ b` that is defeq to `b / a`. |
isOpenMap_div_right (a : G) : IsOpenMap (· / a) := (Homeomorph.divRight a).isOpenMap
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isOpenMap_div_right | null |
isClosedMap_div_right (a : G) : IsClosedMap (· / a) := (Homeomorph.divRight a).isClosedMap
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosedMap_div_right | null |
tendsto_div_nhds_one_iff {α : Type*} {l : Filter α} {x : G} {u : α → G} :
Tendsto (u · / x) l (𝓝 1) ↔ Tendsto u l (𝓝 x) :=
haveI A : Tendsto (fun _ : α => x) l (𝓝 x) := tendsto_const_nhds
⟨fun h => by simpa using h.mul A, fun h => by simpa using h.div' A⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_div_nhds_one_iff | null |
nhds_translation_div (x : G) : comap (· / x) (𝓝 1) = 𝓝 x := by
simpa only [div_eq_mul_inv] using nhds_translation_mul_inv x | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | nhds_translation_div | null |
@[to_additive]
IsTopologicalGroup.t1Space (h : @IsClosed G _ {1}) : T1Space G :=
⟨fun x => by simpa using isClosedMap_mul_right x _ h⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.t1Space | null |
@[to_additive
/-- A subgroup `S` of an additive topological group `G` acts on `G` properly
discontinuously on the left, if it is discrete in the sense that `S ∩ K` is finite for all compact
`K`. (See also `DiscreteTopology`. -/]
Subgroup.properlyDiscontinuousSMul_of_tendsto_cofinite (S : Subgroup G)
(hS : Tendsto S.subtype cofinite (cocompact G)) : ProperlyDiscontinuousSMul S G :=
{ finite_disjoint_inter_image := by
intro K L hK hL
have H : Set.Finite _ := hS ((hL.prod hK).image continuous_div').compl_mem_cocompact
rw [preimage_compl, compl_compl] at H
convert H
ext x
simp only [image_smul, mem_setOf_eq, coe_subtype, mem_preimage, mem_image, Prod.exists]
exact Set.smul_inter_ne_empty_iff' } | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.properlyDiscontinuousSMul_of_tendsto_cofinite | A subgroup `S` of a topological group `G` acts on `G` properly discontinuously on the left, if
it is discrete in the sense that `S ∩ K` is finite for all compact `K`. (See also
`DiscreteTopology`.) |
@[to_additive
/-- A subgroup `S` of an additive topological group `G` acts on `G` properly discontinuously
on the right, if it is discrete in the sense that `S ∩ K` is finite for all compact `K`.
(See also `DiscreteTopology`.)
If `G` is Hausdorff, this can be combined with `t2Space_of_properlyDiscontinuousVAdd_of_t2Space`
to show that the quotient group `G ⧸ S` is Hausdorff. -/]
Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite (S : Subgroup G)
(hS : Tendsto S.subtype cofinite (cocompact G)) : ProperlyDiscontinuousSMul S.op G :=
{ finite_disjoint_inter_image := by
intro K L hK hL
have : Continuous fun p : G × G => (p.1⁻¹, p.2) := continuous_inv.prodMap continuous_id
have H : Set.Finite _ :=
hS ((hK.prod hL).image (continuous_mul.comp this)).compl_mem_cocompact
simp only [preimage_compl, compl_compl, coe_subtype, comp_apply] at H
apply Finite.of_preimage _ (equivOp S).surjective
convert H using 1
ext x
simp only [image_smul, mem_setOf_eq, mem_preimage, mem_image, Prod.exists]
exact Set.op_smul_inter_ne_empty_iff } | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite | A subgroup `S` of a topological group `G` acts on `G` properly discontinuously on the right, if
it is discrete in the sense that `S ∩ K` is finite for all compact `K`. (See also
`DiscreteTopology`.)
If `G` is Hausdorff, this can be combined with `t2Space_of_properlyDiscontinuousSMul_of_t2Space`
to show that the quotient group `G ⧸ S` is Hausdorff. |
@[to_additive
/-- Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of
`0` such that `K + V ⊆ U`. -/]
compact_open_separated_mul_right {K U : Set G} (hK : IsCompact K) (hU : IsOpen U)
(hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), K * V ⊆ U := by
refine hK.induction_on ?_ ?_ ?_ ?_
· exact ⟨univ, by simp⟩
· rintro s t hst ⟨V, hV, hV'⟩
exact ⟨V, hV, (mul_subset_mul_right hst).trans hV'⟩
· rintro s t ⟨V, V_in, hV'⟩ ⟨W, W_in, hW'⟩
use V ∩ W, inter_mem V_in W_in
rw [union_mul]
exact
union_subset ((mul_subset_mul_left V.inter_subset_left).trans hV')
((mul_subset_mul_left V.inter_subset_right).trans hW')
· intro x hx
have := tendsto_mul (show U ∈ 𝓝 (x * 1) by simpa using hU.mem_nhds (hKU hx))
rw [nhds_prod_eq, mem_map, mem_prod_iff] at this
rcases this with ⟨t, ht, s, hs, h⟩
rw [← image_subset_iff, image_mul_prod] at h
exact ⟨t, mem_nhdsWithin_of_mem_nhds ht, s, hs, h⟩
open MulOpposite | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | compact_open_separated_mul_right | Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of `1`
such that `K * V ⊆ U`. |
@[to_additive
/-- Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of
`0` such that `V + K ⊆ U`. -/]
compact_open_separated_mul_left {K U : Set G} (hK : IsCompact K) (hU : IsOpen U)
(hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), V * K ⊆ U := by
rcases compact_open_separated_mul_right (hK.image continuous_op) (opHomeomorph.isOpenMap U hU)
(image_mono hKU) with
⟨V, hV : V ∈ 𝓝 (op (1 : G)), hV' : op '' K * V ⊆ op '' U⟩
refine ⟨op ⁻¹' V, continuous_op.continuousAt hV, ?_⟩
rwa [← image_preimage_eq V op_surjective, ← image_op_mul, image_subset_iff,
preimage_image_eq _ op_injective] at hV' | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | compact_open_separated_mul_left | Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of `1`
such that `V * K ⊆ U`. |
@[to_additive
/-- A compact set is covered by finitely many left additive translates of a set
with non-empty interior. -/]
compact_covered_by_mul_left_translates {K V : Set G} (hK : IsCompact K)
(hV : (interior V).Nonempty) : ∃ t : Finset G, K ⊆ ⋃ g ∈ t, (g * ·) ⁻¹' V := by
obtain ⟨t, ht⟩ : ∃ t : Finset G, K ⊆ ⋃ x ∈ t, interior ((x * ·) ⁻¹' V) := by
refine
hK.elim_finite_subcover (fun x => interior <| (x * ·) ⁻¹' V) (fun x => isOpen_interior) ?_
obtain ⟨g₀, hg₀⟩ := hV
refine fun g _ => mem_iUnion.2 ⟨g₀ * g⁻¹, ?_⟩
refine preimage_interior_subset_interior_preimage (continuous_const.mul continuous_id) ?_
rwa [mem_preimage, Function.id_def, inv_mul_cancel_right]
exact ⟨t, Subset.trans ht <| iUnion₂_mono fun g _ => interior_subset⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | compact_covered_by_mul_left_translates | A compact set is covered by finitely many left multiplicative translates of a set
with non-empty interior. |
@[to_additive
/-- Given two compact sets in a noncompact additive topological group, there is a
translate of the second one that is disjoint from the first one. -/]
exists_disjoint_smul_of_isCompact [NoncompactSpace G] {K L : Set G} (hK : IsCompact K)
(hL : IsCompact L) : ∃ g : G, Disjoint K (g • L) := by
have A : ¬K * L⁻¹ = univ := (hK.mul hL.inv).ne_univ
obtain ⟨g, hg⟩ : ∃ g, g ∉ K * L⁻¹ := by
contrapose! A
exact eq_univ_iff_forall.2 A
refine ⟨g, ?_⟩
refine disjoint_left.2 fun a ha h'a => hg ?_
rcases h'a with ⟨b, bL, rfl⟩
refine ⟨g * b, ha, b⁻¹, by simpa only [Set.mem_inv, inv_inv] using bL, ?_⟩
simp only [mul_inv_cancel_right] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | exists_disjoint_smul_of_isCompact | Every weakly locally compact separable topological group is σ-compact.
Note: this is not true if we drop the topological group hypothesis. -/
@[to_additive SeparableWeaklyLocallyCompactAddGroup.sigmaCompactSpace
/-- Every weakly locally compact separable topological additive group is σ-compact.
Note: this is not true if we drop the topological group hypothesis. -/]
instance (priority := 100) SeparableWeaklyLocallyCompactGroup.sigmaCompactSpace [SeparableSpace G]
[WeaklyLocallyCompactSpace G] : SigmaCompactSpace G := by
obtain ⟨L, hLc, hL1⟩ := exists_compact_mem_nhds (1 : G)
refine ⟨⟨fun n => (fun x => x * denseSeq G n) ⁻¹' L, ?_, ?_⟩⟩
· intro n
exact (Homeomorph.mulRight _).isCompact_preimage.mpr hLc
· refine iUnion_eq_univ_iff.2 fun x => ?_
obtain ⟨_, ⟨n, rfl⟩, hn⟩ : (range (denseSeq G) ∩ (fun y => x * y) ⁻¹' L).Nonempty := by
rw [← (Homeomorph.mulLeft x).apply_symm_apply 1] at hL1
exact (denseRange_denseSeq G).inter_nhds_nonempty
((Homeomorph.mulLeft x).continuous.continuousAt <| hL1)
exact ⟨n, hn⟩
/-- Given two compact sets in a noncompact topological group, there is a translate of the second
one that is disjoint from the first one. |
@[to_additive]
nhds_mul (x y : G) : 𝓝 (x * y) = 𝓝 x * 𝓝 y :=
calc
𝓝 (x * y) = map (x * ·) (map (· * y) (𝓝 1 * 𝓝 1)) := by simp
_ = map₂ (fun a b => x * (a * b * y)) (𝓝 1) (𝓝 1) := by rw [← map₂_mul, map_map₂, map_map₂]
_ = map₂ (fun a b => x * a * (b * y)) (𝓝 1) (𝓝 1) := by simp only [mul_assoc]
_ = 𝓝 x * 𝓝 y := by
rw [← map_mul_left_nhds_one x, ← map_mul_right_nhds_one y, ← map₂_mul, map₂_map_left,
map₂_map_right] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | nhds_mul | null |
@[to_additive (attr := simps)
/-- On an additive topological group, `𝓝 : G → Filter G` can be promoted to an `AddHom`. -/]
nhdsMulHom : G →ₙ* Filter G where
toFun := 𝓝
map_mul' _ _ := nhds_mul _ _ | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | nhdsMulHom | On a topological group, `𝓝 : G → Filter G` can be promoted to a `MulHom`. |
@[to_additive /-- If `G` is an additive group with topological negation, then it is homeomorphic to
its additive units. -/]
toUnits_homeomorph [Group G] [TopologicalSpace G] [ContinuousInv G] : G ≃ₜ Gˣ where
toEquiv := toUnits.toEquiv
continuous_toFun := Units.continuous_iff.2 ⟨continuous_id, continuous_inv⟩
continuous_invFun := Units.continuous_val
@[to_additive] theorem Units.isEmbedding_val [Group G] [TopologicalSpace G] [ContinuousInv G] :
IsEmbedding (val : Gˣ → G) :=
toUnits_homeomorph.symm.isEmbedding | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | toUnits_homeomorph | If `G` is a group with topological `⁻¹`, then it is homeomorphic to its units. |
Continuous.of_coeHom_comp [Group G] [Monoid H] [TopologicalSpace G] [TopologicalSpace H]
[ContinuousInv G] {f : G →* Hˣ} (hf : Continuous ((Units.coeHom H).comp f)) : Continuous f := by
apply continuous_induced_rng.mpr ?_
refine continuous_prodMk.mpr ⟨hf, ?_⟩
simp_rw [← map_inv]
exact MulOpposite.continuous_op.comp (hf.comp continuous_inv) | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Continuous.of_coeHom_comp | null |
@[to_additive]
range_embedProduct [Monoid α] :
Set.range (embedProduct α) = {p : α × αᵐᵒᵖ | p.1 * unop p.2 = 1 ∧ unop p.2 * p.1 = 1} :=
Set.range_eq_iff _ _ |>.mpr
⟨fun a ↦ ⟨a.mul_inv, a.inv_mul⟩, fun p hp ↦ ⟨⟨p.1, unop p.2, hp.1, hp.2⟩, rfl⟩⟩
variable [Monoid α] [TopologicalSpace α] [Monoid β] [TopologicalSpace β]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | range_embedProduct | null |
@[to_additive]
isClosedEmbedding_embedProduct [T1Space α] [ContinuousMul α] :
IsClosedEmbedding (embedProduct α) where
toIsEmbedding := isEmbedding_embedProduct
isClosed_range := by
rw [range_embedProduct]
refine .inter (isClosed_singleton.preimage ?_) (isClosed_singleton.preimage ?_) <;>
fun_prop
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosedEmbedding_embedProduct | null |
_root_.Submonoid.units_isCompact [T1Space α] [ContinuousMul α] {S : Submonoid α}
(hS : IsCompact (S : Set α)) : IsCompact (S.units : Set αˣ) := by
have : IsCompact (S ×ˢ S.op) := hS.prod (opHomeomorph.isCompact_preimage.mp hS)
exact isClosedEmbedding_embedProduct.isCompact_preimage this | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | _root_.Submonoid.units_isCompact | null |
@[to_additive prodAddUnits
/-- The topological group isomorphism between the additive units of a product of two
additive monoids, and the product of the additive units of each additive monoid. -/]
_root_.Homeomorph.prodUnits : (α × β)ˣ ≃ₜ αˣ × βˣ where
continuous_toFun :=
(continuous_fst.units_map (MonoidHom.fst α β)).prodMk
(continuous_snd.units_map (MonoidHom.snd α β))
continuous_invFun :=
Units.continuous_iff.2
⟨continuous_val.fst'.prodMk continuous_val.snd',
continuous_coe_inv.fst'.prodMk continuous_coe_inv.snd'⟩
toEquiv := MulEquiv.prodUnits.toEquiv | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | _root_.Homeomorph.prodUnits | The topological group isomorphism between the units of a product of two monoids, and the product
of the units of each monoid. |
@[to_additive]
topologicalGroup_sInf {ts : Set (TopologicalSpace G)}
(h : ∀ t ∈ ts, @IsTopologicalGroup G t _) : @IsTopologicalGroup G (sInf ts) _ :=
letI := sInf ts
{ toContinuousInv :=
@continuousInv_sInf _ _ _ fun t ht => @IsTopologicalGroup.toContinuousInv G t _ <| h t ht
toContinuousMul :=
@continuousMul_sInf _ _ _ fun t ht =>
@IsTopologicalGroup.toContinuousMul G t _ <| h t ht }
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | topologicalGroup_sInf | null |
topologicalGroup_iInf {ts' : ι → TopologicalSpace G}
(h' : ∀ i, @IsTopologicalGroup G (ts' i) _) : @IsTopologicalGroup G (⨅ i, ts' i) _ := by
rw [← sInf_range]
exact topologicalGroup_sInf (Set.forall_mem_range.mpr h')
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | topologicalGroup_iInf | null |
topologicalGroup_inf {t₁ t₂ : TopologicalSpace G} (h₁ : @IsTopologicalGroup G t₁ _)
(h₂ : @IsTopologicalGroup G t₂ _) : @IsTopologicalGroup G (t₁ ⊓ t₂) _ := by
rw [inf_eq_iInf]
refine topologicalGroup_iInf fun b => ?_
cases b <;> assumption | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | topologicalGroup_inf | null |
@[ext]
ClosedSubgroup (G : Type u) [Group G] [TopologicalSpace G] extends Subgroup G where
isClosed' : IsClosed carrier | structure | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | ClosedSubgroup | The type of closed subgroups of a topological group. |
@[ext]
ClosedAddSubgroup (G : Type u) [AddGroup G] [TopologicalSpace G] extends
AddSubgroup G where
isClosed' : IsClosed carrier
attribute [to_additive] ClosedSubgroup
attribute [coe] ClosedSubgroup.toSubgroup ClosedAddSubgroup.toAddSubgroup | structure | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | ClosedAddSubgroup | The type of closed subgroups of an additive topological group. |
@[to_additive]
toSubgroup_injective : Function.Injective
(ClosedSubgroup.toSubgroup : ClosedSubgroup G → Subgroup G) :=
fun A B h ↦ by
ext
rw [h]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | toSubgroup_injective | null |
@[to_additive]
instInfClosedSubgroup : Min (ClosedSubgroup G) :=
⟨fun U V ↦ ⟨U ⊓ V, U.isClosed'.inter V.isClosed'⟩⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | instInfClosedSubgroup | null |
instSemilatticeInfClosedSubgroup : SemilatticeInf (ClosedSubgroup G) :=
SetLike.coe_injective.semilatticeInf ((↑) : ClosedSubgroup G → Set G) fun _ _ ↦ rfl
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | instSemilatticeInfClosedSubgroup | null |
normalCore_isClosed (H : Subgroup G) (h : IsClosed (H : Set G)) :
IsClosed (H.normalCore : Set G) := by
rw [normalCore_eq_iInf_conjAct]
push_cast
apply isClosed_iInter
intro g
convert IsClosed.preimage (IsTopologicalGroup.continuous_conj (ConjAct.ofConjAct g⁻¹)) h using 1
exact Set.ext (fun t ↦ Set.mem_smul_set_iff_inv_smul_mem)
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | normalCore_isClosed | null |
isOpen_of_isClosed_of_finiteIndex (H : Subgroup G) [H.FiniteIndex]
(h : IsClosed (H : Set G)) : IsOpen (H : Set G) := by
apply isClosed_compl_iff.mp
convert isClosed_iUnion_of_finite <| fun (x : {x : (G ⧸ H) // x ≠ QuotientGroup.mk 1})
↦ IsClosed.smul h (Quotient.out x.1)
ext x
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· have : QuotientGroup.mk 1 ≠ QuotientGroup.mk (s := H) x := by
apply QuotientGroup.eq.not.mpr
simpa only [inv_one, one_mul, ne_eq]
simp only [ne_eq, Set.mem_iUnion]
use ⟨QuotientGroup.mk (s := H) x, this.symm⟩,
(Quotient.out (QuotientGroup.mk (s := H) x))⁻¹ * x
simp only [SetLike.mem_coe, smul_eq_mul, mul_inv_cancel_left, and_true]
exact QuotientGroup.eq.mp <| QuotientGroup.out_eq' (QuotientGroup.mk (s := H) x)
· rcases h with ⟨S,⟨y,hS⟩,mem⟩
simp only [← hS] at mem
rcases mem with ⟨h,hh,eq⟩
simp only [Set.mem_compl_iff, SetLike.mem_coe]
by_contra mH
simp only [← eq, ne_eq, smul_eq_mul] at mH
absurd y.2.symm
rw [← QuotientGroup.out_eq' y.1, QuotientGroup.eq]
simp only [inv_one, ne_eq, one_mul, (Subgroup.mul_mem_cancel_right H hh).mp mH] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | isOpen_of_isClosed_of_finiteIndex | null |
@[to_additive
/-- Every topological additive group
in which there exists a compact set with nonempty interior is locally compact. -/]
TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
(K : PositiveCompacts G) : LocallyCompactSpace G :=
let ⟨_x, hx⟩ := K.interior_nonempty
K.isCompact.locallyCompactSpace_of_mem_nhds_of_group (mem_interior_iff_mem_nhds.1 hx) | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/Algebra/Group/Compact.lean | TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group | Every topological group in which there exists a compact set with nonempty interior
is locally compact. |
@[to_additive]
isInducing_toContinuousMap :
IsInducing (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) := ⟨rfl⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | isInducing_toContinuousMap | null |
isEmbedding_toContinuousMap :
IsEmbedding (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) :=
⟨isInducing_toContinuousMap A B, toContinuousMap_injective⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | isEmbedding_toContinuousMap | null |
instContinuousEvalConst : ContinuousEvalConst (ContinuousMonoidHom A B) A B :=
.of_continuous_forget (isInducing_toContinuousMap A B).continuous
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | instContinuousEvalConst | null |
instContinuousEval [LocallyCompactPair A B] :
ContinuousEval (ContinuousMonoidHom A B) A B :=
.of_continuous_forget (isInducing_toContinuousMap A B).continuous
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | instContinuousEval | null |
range_toContinuousMap :
Set.range (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) =
{f : C(A, B) | f 1 = 1 ∧ ∀ x y, f (x * y) = f x * f y} := by
refine Set.Subset.antisymm (Set.range_subset_iff.2 fun f ↦ ⟨map_one f, map_mul f⟩) ?_
rintro f ⟨h1, hmul⟩
exact ⟨{ f with map_one' := h1, map_mul' := hmul }, rfl⟩
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | range_toContinuousMap | null |
isClosedEmbedding_toContinuousMap [ContinuousMul B] [T2Space B] :
IsClosedEmbedding (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) where
toIsEmbedding := isEmbedding_toContinuousMap A B
isClosed_range := by
simp only [range_toContinuousMap, Set.setOf_and, Set.setOf_forall]
refine .inter (isClosed_singleton.preimage (continuous_eval_const 1)) <|
isClosed_iInter fun x ↦ isClosed_iInter fun y ↦ ?_
exact isClosed_eq (continuous_eval_const (x * y)) <|
.mul (continuous_eval_const x) (continuous_eval_const y)
variable {A B C D E}
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | isClosedEmbedding_toContinuousMap | null |
@[to_additive]
continuous_of_continuous_uncurry {A : Type*} [TopologicalSpace A]
(f : A → ContinuousMonoidHom B C) (h : Continuous (Function.uncurry fun x y => f x y)) :
Continuous f :=
(isInducing_toContinuousMap _ _).continuous_iff.mpr
(ContinuousMap.continuous_of_continuous_uncurry _ h)
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | continuous_of_continuous_uncurry | null |
continuous_comp [LocallyCompactSpace B] :
Continuous fun f : ContinuousMonoidHom A B × ContinuousMonoidHom B C => f.2.comp f.1 :=
(isInducing_toContinuousMap A C).continuous_iff.2 <|
ContinuousMap.continuous_comp'.comp
((isInducing_toContinuousMap A B).prodMap (isInducing_toContinuousMap B C)).continuous
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | continuous_comp | null |
continuous_comp_left (f : ContinuousMonoidHom A B) :
Continuous fun g : ContinuousMonoidHom B C => g.comp f :=
(isInducing_toContinuousMap A C).continuous_iff.2 <|
f.toContinuousMap.continuous_precomp.comp (isInducing_toContinuousMap B C).continuous
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | continuous_comp_left | null |
continuous_comp_right (f : ContinuousMonoidHom B C) :
Continuous fun g : ContinuousMonoidHom A B => f.comp g :=
(isInducing_toContinuousMap A C).continuous_iff.2 <|
f.toContinuousMap.continuous_postcomp.comp (isInducing_toContinuousMap A B).continuous
variable (E) in | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | continuous_comp_right | null |
@[to_additive /-- `ContinuousAddMonoidHom _ f` is a functor. -/]
compLeft (f : ContinuousMonoidHom A B) :
ContinuousMonoidHom (ContinuousMonoidHom B E) (ContinuousMonoidHom A E) where
toFun g := g.comp f
map_one' := rfl
map_mul' _g _h := rfl
continuous_toFun := f.continuous_comp_left
variable (A) in | def | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | compLeft | `ContinuousMonoidHom _ f` is a functor. |
@[to_additive /-- `ContinuousAddMonoidHom f _` is a functor. -/]
compRight {B : Type*} [CommGroup B] [TopologicalSpace B] [IsTopologicalGroup B]
(f : ContinuousMonoidHom B E) :
ContinuousMonoidHom (ContinuousMonoidHom A B) (ContinuousMonoidHom A E) where
toFun g := f.comp g
map_one' := ext fun _a => map_one f
map_mul' g h := ext fun a => map_mul f (g a) (h a)
continuous_toFun := f.continuous_comp_right | def | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | compRight | `ContinuousMonoidHom f _` is a functor. |
@[to_additive]
isClosedEmbedding_coe : IsClosedEmbedding ((⇑) : (A →ₜ* B) → A → B) :=
ContinuousMap.isHomeomorph_coe.isClosedEmbedding.comp <| isClosedEmbedding_toContinuousMap ..
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | isClosedEmbedding_coe | null |
@[to_additive]
locallyCompactSpace_of_equicontinuousAt (U : Set X) (V : Set Y)
(hU : IsCompact U) (hV : V ∈ nhds (1 : Y))
(h : EquicontinuousAt (fun f : {f : X →* Y | Set.MapsTo f U V} ↦ (f : X → Y)) 1) :
LocallyCompactSpace (ContinuousMonoidHom X Y) := by
replace h := equicontinuous_of_equicontinuousAt_one _ h
obtain ⟨W, hWo, hWV, hWc⟩ := local_compact_nhds hV
let S1 : Set (X →* Y) := {f | Set.MapsTo f U W}
let S2 : Set (ContinuousMonoidHom X Y) := {f | Set.MapsTo f U W}
let S3 : Set C(X, Y) := (↑) '' S2
let S4 : Set (X → Y) := (↑) '' S3
replace h : Equicontinuous ((↑) : S1 → X → Y) :=
h.comp (Subtype.map _root_.id fun f hf ↦ hf.mono_right hWV)
have hS4 : S4 = (↑) '' S1 := by
ext
constructor
· rintro ⟨-, ⟨f, hf, rfl⟩, rfl⟩
exact ⟨f, hf, rfl⟩
· rintro ⟨f, hf, rfl⟩
exact ⟨⟨f, h.continuous ⟨f, hf⟩⟩, ⟨⟨f, h.continuous ⟨f, hf⟩⟩, hf, rfl⟩, rfl⟩
replace h : Equicontinuous ((↑) : S3 → X → Y) := by
rw [equicontinuous_iff_range, ← Set.image_eq_range] at h ⊢
rwa [← hS4] at h
replace hS4 : S4 = Set.pi U (fun _ ↦ W) ∩ Set.range ((↑) : (X →* Y) → (X → Y)) := by
simp_rw [hS4, Set.ext_iff, Set.mem_image, S1, Set.mem_setOf_eq]
exact fun f ↦ ⟨fun ⟨g, hg, hf⟩ ↦ hf ▸ ⟨hg, g, rfl⟩, fun ⟨hg, g, hf⟩ ↦ ⟨g, hf ▸ hg, hf⟩⟩
replace hS4 : IsClosed S4 :=
hS4.symm ▸ (isClosed_set_pi (fun _ _ ↦ hWc.isClosed)).inter (MonoidHom.isClosed_range_coe X Y)
have hS2 : (interior S2).Nonempty := by
let T : Set (ContinuousMonoidHom X Y) := {f | Set.MapsTo f U (interior W)}
have h1 : T.Nonempty := ⟨1, fun _ _ ↦ mem_interior_iff_mem_nhds.mpr hWo⟩
have h2 : T ⊆ S2 := fun f hf ↦ hf.mono_right interior_subset
have h3 : IsOpen T := isOpen_induced (ContinuousMap.isOpen_setOf_mapsTo hU isOpen_interior)
exact h1.mono (interior_maximal h2 h3)
exact TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
⟨⟨S2, (isInducing_toContinuousMap X Y).isCompact_iff.mpr
(ArzelaAscoli.isCompact_of_equicontinuous S3 hS4.isCompact h)⟩, hS2⟩
variable [LocallyCompactSpace X]
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | locallyCompactSpace_of_equicontinuousAt | null |
locallyCompactSpace_of_hasBasis (V : ℕ → Set Y)
(hV : ∀ {n x}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1))
(hVo : Filter.HasBasis (nhds 1) (fun _ ↦ True) V) :
LocallyCompactSpace (ContinuousMonoidHom X Y) := by
obtain ⟨U0, hU0c, hU0o⟩ := exists_compact_mem_nhds (1 : X)
let U_aux : ℕ → {S : Set X | S ∈ nhds 1} :=
Nat.rec ⟨U0, hU0o⟩ <| fun _ S ↦ let h := exists_closed_nhds_one_inv_eq_mul_subset S.2
⟨Classical.choose h, (Classical.choose_spec h).1⟩
let U : ℕ → Set X := fun n ↦ (U_aux n).1
have hU1 : ∀ n, U n ∈ nhds 1 := fun n ↦ (U_aux n).2
have hU2 : ∀ n, U (n + 1) * U (n + 1) ⊆ U n :=
fun n ↦ (Classical.choose_spec (exists_closed_nhds_one_inv_eq_mul_subset (U_aux n).2)).2.2.2
have hU3 : ∀ n, U (n + 1) ⊆ U n :=
fun n x hx ↦ hU2 n (mul_one x ▸ Set.mul_mem_mul hx (mem_of_mem_nhds (hU1 (n + 1))))
have hU4 : ∀ f : X →* Y, Set.MapsTo f (U 0) (V 0) → ∀ n, Set.MapsTo f (U n) (V n) := by
intro f hf n
induction n with
| zero => exact hf
| succ n ih =>
exact fun x hx ↦ hV (ih (hU3 n hx)) (map_mul f x x ▸ ih (hU2 n (Set.mul_mem_mul hx hx)))
apply locallyCompactSpace_of_equicontinuousAt (U 0) (V 0) hU0c (hVo.mem_of_mem trivial)
rw [hVo.uniformity_of_nhds_one.equicontinuousAt_iff_right]
refine fun n _ ↦ Filter.eventually_iff_exists_mem.mpr ⟨U n, hU1 n, fun x hx ⟨f, hf⟩ ↦ ?_⟩
rw [Set.mem_setOf_eq, map_one, div_one]
exact hU4 f hf n hx | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | locallyCompactSpace_of_hasBasis | null |
ContinuousNeg (G : Type u) [TopologicalSpace G] [Neg G] : Prop where
continuous_neg : Continuous fun a : G => -a
attribute [continuity, fun_prop] ContinuousNeg.continuous_neg | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousNeg | Basic hypothesis to talk about a topological additive group. A topological additive group
over `M`, for example, is obtained by requiring the instances `AddGroup M` and
`ContinuousAdd M` and `ContinuousNeg M`. |
@[to_additive (attr := continuity)]
ContinuousInv (G : Type u) [TopologicalSpace G] [Inv G] : Prop where
continuous_inv : Continuous fun a : G => a⁻¹
attribute [continuity, fun_prop] ContinuousInv.continuous_inv
export ContinuousInv (continuous_inv)
export ContinuousNeg (continuous_neg) | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousInv | Basic hypothesis to talk about a topological group. A topological group over `M`, for example,
is obtained by requiring the instances `Group M` and `ContinuousMul M` and
`ContinuousInv M`. |
@[to_additive
/-- If a function converges to a value in an additive topological group, then its
negation converges to the negation of this value. -/]
Filter.Tendsto.inv {f : α → G} {l : Filter α} {y : G} (h : Tendsto f l (𝓝 y)) :
Tendsto (fun x => (f x)⁻¹) l (𝓝 y⁻¹) :=
(continuous_inv.tendsto y).comp h
variable {f : X → G} {s : Set X} {x : X}
@[to_additive (attr := continuity, fun_prop)] | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | Filter.Tendsto.inv | If a function converges to a value in a multiplicative topological group, then its inverse
converges to the inverse of this value.
For the version in topological groups with zero (including topological fields)
assuming additionally that the limit is nonzero, use `Filter.Tendsto.inv₀`. |
Continuous.inv (hf : Continuous f) : Continuous fun x => (f x)⁻¹ :=
continuous_inv.comp hf
@[to_additive]
nonrec theorem ContinuousWithinAt.inv (hf : ContinuousWithinAt f s x) :
ContinuousWithinAt (fun x => (f x)⁻¹) s x :=
hf.inv
@[to_additive (attr := fun_prop)]
nonrec theorem ContinuousAt.inv (hf : ContinuousAt f x) : ContinuousAt (fun x => (f x)⁻¹) x :=
hf.inv
@[to_additive (attr := fun_prop)] | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | Continuous.inv | null |
ContinuousOn.inv (hf : ContinuousOn f s) : ContinuousOn (fun x => (f x)⁻¹) s := fun x hx ↦
(hf x hx).inv | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousOn.inv | null |
IsTopologicalAddGroup (G : Type u) [TopologicalSpace G] [AddGroup G] : Prop
extends ContinuousAdd G, ContinuousNeg G | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | IsTopologicalAddGroup | A topological (additive) group is a group in which the addition and negation operations are
continuous.
When you declare an instance that does not already have a `UniformSpace` instance,
you should also provide an instance of `UniformSpace` and `IsUniformAddGroup` using
`IsTopologicalAddGroup.toUniformSpace` and `isUniformAddGroup_of_addCommGroup`. |
@[to_additive]
IsTopologicalGroup (G : Type*) [TopologicalSpace G] [Group G] : Prop
extends ContinuousMul G, ContinuousInv G | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | IsTopologicalGroup | A topological group is a group in which the multiplication and inversion operations are
continuous.
When you declare an instance that does not already have a `UniformSpace` instance,
you should also provide an instance of `UniformSpace` and `IsUniformGroup` using
`IsTopologicalGroup.toUniformSpace` and `isUniformGroup_of_commGroup`. |
ContinuousSub (G : Type*) [TopologicalSpace G] [Sub G] : Prop where
continuous_sub : Continuous fun p : G × G => p.1 - p.2 | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousSub | A typeclass saying that `p : G × G ↦ p.1 - p.2` is a continuous function. This property
automatically holds for topological additive groups but it also holds, e.g., for `ℝ≥0`. |
@[to_additive existing]
ContinuousDiv (G : Type*) [TopologicalSpace G] [Div G] : Prop where
continuous_div' : Continuous fun p : G × G => p.1 / p.2
@[to_additive] | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousDiv | A typeclass saying that `p : G × G ↦ p.1 / p.2` is a continuous function. This property
automatically holds for topological groups. Lemmas using this class have primes.
The unprimed version is for `GroupWithZero`. |
@[to_additive sub]
Filter.Tendsto.div' {f g : α → G} {l : Filter α} {a b : G} (hf : Tendsto f l (𝓝 a))
(hg : Tendsto g l (𝓝 b)) : Tendsto (fun x => f x / g x) l (𝓝 (a / b)) :=
(continuous_div'.tendsto (a, b)).comp (hf.prodMk_nhds hg)
variable {f g : X → G} {s : Set X} {x : X}
@[to_additive (attr := fun_prop) sub]
nonrec theorem ContinuousAt.div' (hf : ContinuousAt f x) (hg : ContinuousAt g x) :
ContinuousAt (fun x => f x / g x) x :=
hf.div' hg
@[to_additive sub] | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | Filter.Tendsto.div' | null |
ContinuousWithinAt.div' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (fun x => f x / g x) s x :=
Filter.Tendsto.div' hf hg
@[to_additive (attr := fun_prop) sub] | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousWithinAt.div' | null |
ContinuousOn.div' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun x => f x / g x) s := fun x hx => (hf x hx).div' (hg x hx)
@[to_additive (attr := continuity, fun_prop) sub] | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousOn.div' | null |
Continuous.div' (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x / g x :=
continuous_div'.comp₂ hf hg | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | Continuous.div' | null |
GroupTopology (α : Type u) [Group α] : Type u
extends TopologicalSpace α, IsTopologicalGroup α | structure | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | GroupTopology | A group topology on a group `α` is a topology for which multiplication and inversion
are continuous. |
AddGroupTopology (α : Type u) [AddGroup α] : Type u
extends TopologicalSpace α, IsTopologicalAddGroup α
attribute [to_additive] GroupTopology | structure | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | AddGroupTopology | An additive group topology on an additive group `α` is a topology for which addition and
negation are continuous. |
@[to_additive /-- A version of the global `continuous_add` suitable for dot notation. -/]
continuous_mul' (g : GroupTopology α) :
haveI := g.toTopologicalSpace
Continuous fun p : α × α => p.1 * p.2 := by
letI := g.toTopologicalSpace
haveI := g.toIsTopologicalGroup
exact continuous_mul | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | continuous_mul' | A version of the global `continuous_mul` suitable for dot notation. |
@[to_additive /-- A version of the global `continuous_neg` suitable for dot notation. -/]
continuous_inv' (g : GroupTopology α) :
haveI := g.toTopologicalSpace
Continuous (Inv.inv : α → α) := by
letI := g.toTopologicalSpace
haveI := g.toIsTopologicalGroup
exact continuous_inv
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | continuous_inv' | A version of the global `continuous_inv` suitable for dot notation. |
toTopologicalSpace_injective :
Function.Injective (toTopologicalSpace : GroupTopology α → TopologicalSpace α) :=
fun f g h => by
cases f
cases g
congr
@[to_additive (attr := ext)] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | toTopologicalSpace_injective | null |
ext' {f g : GroupTopology α} (h : f.IsOpen = g.IsOpen) : f = g :=
toTopologicalSpace_injective <| TopologicalSpace.ext h | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | ext' | null |
@[to_additive
/-- Given `f : α → β` and a topology on `α`, the coinduced additive group topology on `β`
is the finest topology such that `f` is continuous and `β` is a topological additive group. -/]
coinduced {α β : Type*} [t : TopologicalSpace α] [Group β] (f : α → β) : GroupTopology β :=
sInf { b : GroupTopology β | TopologicalSpace.coinduced f t ≤ b.toTopologicalSpace }
@[to_additive] | def | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | coinduced | The ordering on group topologies on the group `γ`. `t ≤ s` if every set open in `s` is also open
in `t` (`t` is finer than `s`). -/
@[to_additive
/-- The ordering on group topologies on the group `γ`. `t ≤ s` if every set open in `s`
is also open in `t` (`t` is finer than `s`). -/]
instance : PartialOrder (GroupTopology α) :=
PartialOrder.lift toTopologicalSpace toTopologicalSpace_injective
@[to_additive (attr := simp)]
theorem toTopologicalSpace_le {x y : GroupTopology α} :
x.toTopologicalSpace ≤ y.toTopologicalSpace ↔ x ≤ y :=
Iff.rfl
@[to_additive]
instance : Top (GroupTopology α) :=
let _t : TopologicalSpace α := ⊤
⟨{ continuous_mul := continuous_top
continuous_inv := continuous_top }⟩
@[to_additive (attr := simp)]
theorem toTopologicalSpace_top : (⊤ : GroupTopology α).toTopologicalSpace = ⊤ :=
rfl
@[to_additive]
instance : Bot (GroupTopology α) :=
let _t : TopologicalSpace α := ⊥
⟨{ continuous_mul := by
haveI := discreteTopology_bot α
fun_prop
continuous_inv := continuous_bot }⟩
@[to_additive (attr := simp)]
theorem toTopologicalSpace_bot : (⊥ : GroupTopology α).toTopologicalSpace = ⊥ :=
rfl
@[to_additive]
instance : BoundedOrder (GroupTopology α) where
top := ⊤
le_top x := show x.toTopologicalSpace ≤ ⊤ from le_top
bot := ⊥
bot_le x := show ⊥ ≤ x.toTopologicalSpace from bot_le
@[to_additive]
instance : Min (GroupTopology α) where min x y := ⟨x.1 ⊓ y.1, topologicalGroup_inf x.2 y.2⟩
@[to_additive (attr := simp)]
theorem toTopologicalSpace_inf (x y : GroupTopology α) :
(x ⊓ y).toTopologicalSpace = x.toTopologicalSpace ⊓ y.toTopologicalSpace :=
rfl
@[to_additive]
instance : SemilatticeInf (GroupTopology α) :=
toTopologicalSpace_injective.semilatticeInf _ toTopologicalSpace_inf
@[to_additive]
instance : Inhabited (GroupTopology α) :=
⟨⊤⟩
/-- Infimum of a collection of group topologies. -/
@[to_additive /-- Infimum of a collection of additive group topologies -/]
instance : InfSet (GroupTopology α) where
sInf S :=
⟨sInf (toTopologicalSpace '' S), topologicalGroup_sInf <| forall_mem_image.2 fun t _ => t.2⟩
@[to_additive (attr := simp)]
theorem toTopologicalSpace_sInf (s : Set (GroupTopology α)) :
(sInf s).toTopologicalSpace = sInf (toTopologicalSpace '' s) := rfl
@[to_additive (attr := simp)]
theorem toTopologicalSpace_iInf {ι} (s : ι → GroupTopology α) :
(⨅ i, s i).toTopologicalSpace = ⨅ i, (s i).toTopologicalSpace :=
congr_arg sInf (range_comp _ _).symm
/-- Group topologies on `γ` form a complete lattice, with `⊥` the discrete topology and `⊤` the
indiscrete topology.
The infimum of a collection of group topologies is the topology generated by all their open sets
(which is a group topology).
The supremum of two group topologies `s` and `t` is the infimum of the family of all group
topologies contained in the intersection of `s` and `t`. -/
@[to_additive
/-- Group topologies on `γ` form a complete lattice, with `⊥` the discrete topology and
`⊤` the indiscrete topology.
The infimum of a collection of group topologies is the topology generated by all their open sets
(which is a group topology).
The supremum of two group topologies `s` and `t` is the infimum of the family of all group
topologies contained in the intersection of `s` and `t`. -/]
instance : CompleteSemilatticeInf (GroupTopology α) :=
{ inferInstanceAs (InfSet (GroupTopology α)),
inferInstanceAs (PartialOrder (GroupTopology α)) with
sInf_le := fun _ a haS => toTopologicalSpace_le.1 <| sInf_le ⟨a, haS, rfl⟩
le_sInf := by
intro S a hab
apply (inferInstanceAs (CompleteLattice (TopologicalSpace α))).le_sInf
rintro _ ⟨b, hbS, rfl⟩
exact hab b hbS }
@[to_additive]
instance : CompleteLattice (GroupTopology α) :=
{ inferInstanceAs (BoundedOrder (GroupTopology α)),
inferInstanceAs (SemilatticeInf (GroupTopology α)),
completeLatticeOfCompleteSemilatticeInf _ with
inf := (· ⊓ ·)
top := ⊤
bot := ⊥ }
/-- Given `f : α → β` and a topology on `α`, the coinduced group topology on `β` is the finest
topology such that `f` is continuous and `β` is a topological group. |
coinduced_continuous {α β : Type*} [t : TopologicalSpace α] [Group β] (f : α → β) :
Continuous[t, (coinduced f).toTopologicalSpace] f := by
rw [continuous_sInf_rng]
rintro _ ⟨t', ht', rfl⟩
exact continuous_iff_coinduced_le.2 ht' | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | coinduced_continuous | null |
@[to_additive /-- Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point. -/]
smul_singleton_mem_nhds_of_sigmaCompact
{U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
/- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
if `V` is small enough. -/
obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
exists_closed_nhds_one_inv_eq_mul_subset hU
obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ :=
countable_cover_nhds_of_sigmaCompact fun _ ↦ by simpa
let K : ℕ → Set G := compactCovering G
let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
have : Nonempty X := ⟨x⟩
have : Encodable s := Countable.toEncodable s_count
apply nonempty_interior_of_iUnion_of_closed
· rintro ⟨n, ⟨g, hg⟩⟩
apply IsCompact.isClosed
suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
simpa only [F, smul_singleton] using H
apply IsCompact.image
· exact (isCompact_compactCovering G n).inter_right (V_closed.smul g)
· exact continuous_id.smul continuous_const
· apply eq_univ_iff_forall.2 (fun y ↦ ?_)
obtain ⟨h, rfl⟩ : ∃ h, h • x = y := exists_smul_eq G x y
obtain ⟨n, hn⟩ : ∃ n, h ∈ K n := exists_mem_compactCovering h
obtain ⟨g, gs, hg⟩ : ∃ g ∈ s, h ∈ g • V := exists_set_mem_of_union_eq_top s _ hs _
simp only [F, smul_singleton, mem_iUnion, mem_image, mem_inter_iff, Prod.exists,
Subtype.exists, exists_prop]
exact ⟨n, g, gs, h, ⟨hn, hg⟩, rfl⟩
have I : (interior ((g • V) • {x})).Nonempty := by
apply hi.mono
apply interior_mono
exact smul_subset_smul_right inter_subset_right
obtain ⟨y, hy⟩ : (interior (V • ({x} : Set X))).Nonempty := by
rw [smul_assoc, interior_smul] at I
exact smul_set_nonempty.1 I
obtain ⟨g', hg', rfl⟩ : ∃ g' ∈ V, g' • x = y := by simpa using interior_subset hy
have J : (g'⁻¹ • V) • {x} ∈ 𝓝 x := by
apply mem_interior_iff_mem_nhds.1
rwa [smul_assoc, interior_smul, mem_inv_smul_set_iff]
have : (g'⁻¹ • V) • {x} ⊆ U • ({x} : Set X) := by
apply smul_subset_smul_right
apply Subset.trans (smul_set_subset_smul (inv_mem_inv.2 hg')) ?_
rw [V_symm]
... | theorem | Topology | [
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/OpenMapping.lean | smul_singleton_mem_nhds_of_sigmaCompact | Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. |
@[to_additive /-- Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open. This is a version of the open mapping theorem, valid
notably for the action of a sigma-compact locally compact group on a locally compact space. -/]
isOpenMap_smul_of_sigmaCompact (x : X) : IsOpenMap (fun (g : G) ↦ g • x) := by
/- We have already proved the theorem around the basepoint of the orbit, in
`smul_singleton_mem_nhds_of_sigmaCompact`. The general statement follows around an arbitrary
point by changing basepoints. -/
simp_rw [isOpenMap_iff_nhds_le, Filter.le_map_iff]
intro g U hU
have : (· • x) = (· • (g • x)) ∘ (· * g⁻¹) := by
ext g
simp [smul_smul]
rw [this, image_comp, ← smul_singleton]
apply smul_singleton_mem_nhds_of_sigmaCompact
simpa using isOpenMap_mul_right g⁻¹ |>.image_mem_nhds hU | theorem | Topology | [
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/OpenMapping.lean | isOpenMap_smul_of_sigmaCompact | Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open. This is a version of the open mapping theorem, valid notably for the
action of a sigma-compact locally compact group on a locally compact space. |
@[to_additive]
MonoidHom.isOpenMap_of_sigmaCompact
{H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]
(f : G →* H) (hf : Function.Surjective f) (h'f : Continuous f) :
IsOpenMap f := by
let A : MulAction G H := MulAction.compHom _ f
have : ContinuousSMul G H := continuousSMul_compHom h'f
have : IsPretransitive G H := isPretransitive_compHom hf
have : f = (fun (g : G) ↦ g • (1 : H)) := by simp [A, MulAction.compHom_smul_def]
rw [this]
exact isOpenMap_smul_of_sigmaCompact _ | theorem | Topology | [
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/OpenMapping.lean | MonoidHom.isOpenMap_of_sigmaCompact | A surjective morphism of topological groups is open when the source group is sigma-compact and
the target group is a Baire space (for instance a locally compact group). |
@[to_additive]
subset_interior_smul : interior s • interior t ⊆ interior (s • t) :=
(Set.smul_subset_smul_right interior_subset).trans subset_interior_smul_right | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_smul | null |
@[to_additive]
IsClosed.smul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
IsClosed (s • t) := by
have : ∀ x ∈ s • t, ∃ g ∈ s, g⁻¹ • x ∈ t := by
rintro x ⟨g, hgs, y, hyt, rfl⟩
refine ⟨g, hgs, ?_⟩
rwa [inv_smul_smul]
choose! f hf using this
refine isClosed_of_closure_subset (fun x hx ↦ ?_)
rcases mem_closure_iff_ultrafilter.mp hx with ⟨u, hust, hux⟩
have : Ultrafilter.map f u ≤ 𝓟 s :=
calc Ultrafilter.map f u ≤ map f (𝓟 (s • t)) := map_mono (le_principal_iff.mpr hust)
_ = 𝓟 (f '' (s • t)) := map_principal
_ ≤ 𝓟 s := principal_mono.mpr (image_subset_iff.mpr (fun x hx ↦ (hf x hx).1))
rcases hs.ultrafilter_le_nhds (Ultrafilter.map f u) this with ⟨g, hg, hug⟩
suffices g⁻¹ • x ∈ t from
⟨g, hg, g⁻¹ • x, this, smul_inv_smul _ _⟩
exact ht.mem_of_tendsto ((Tendsto.inv hug).smul hux)
(Eventually.mono hust (fun y hy ↦ (hf y hy).2))
/-! One may expect a version of `IsClosed.smul_left_of_isCompact` where `t` is compact and `s` is
closed, but such a lemma can't be true in this level of generality. For a counterexample, consider
`ℚ` acting on `ℝ` by translation, and let `s : Set ℚ := univ`, `t : set ℝ := {0}`. Then `s` is
closed and `t` is compact, but `s +ᵥ t` is the set of all rationals, which is definitely not
closed in `ℝ`.
To fix the proof, we would need to make two additional assumptions:
- for any `x ∈ t`, `s • {x}` is closed
- for any `x ∈ t`, there is a continuous function `g : s • {x} → s` such that, for all
`y ∈ s • {x}`, we have `y = (g y) • x`
These are fairly specific hypotheses so we don't state this version of the lemmas, but an
interesting fact is that these two assumptions are verified in the case of an
`IsTopologicalAddTorsor`. We prove this special case in `IsClosed.vadd_right_of_isCompact`. -/
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsClosed.smul_left_of_isCompact | null |
MulAction.isClosedMap_quotient [CompactSpace α] :
letI := orbitRel α β
IsClosedMap (Quotient.mk' : β → Quotient (orbitRel α β)) := by
intro t ht
rw [← isQuotientMap_quotient_mk'.isClosed_preimage,
MulAction.quotient_preimage_image_eq_union_mul]
convert ht.smul_left_of_isCompact (isCompact_univ (X := α))
rw [← biUnion_univ, ← iUnion_smul_left_image]
simp only [image_smul] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | MulAction.isClosedMap_quotient | null |
@[to_additive]
IsOpen.mul_left : IsOpen t → IsOpen (s * t) :=
IsOpen.smul_left
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.mul_left | null |
subset_interior_mul_right : s * interior t ⊆ interior (s * t) :=
subset_interior_smul_right
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_mul_right | null |
subset_interior_mul : interior s * interior t ⊆ interior (s * t) :=
subset_interior_smul
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_mul | null |
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